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The double commutant property for composition operators
| Content Provider | Semantic Scholar |
|---|---|
| Author | Lacruz, Miguel León-Saavedra, Fernando Srdjan P. Petrović Rodríguez-Piazza, Luis |
| Copyright Year | 2019 |
| Abstract | We investigate the double commutant property for a composition operator $$C_\varphi $$Cφ, induced on the Hardy space $$H^2({\mathbb {D}})$$H2(D) by a linear fractional self-map $$\varphi $$φ of the unit disk $${\mathbb {D}}.$$D. Our main result is that this property always holds, except when $$\varphi $$φ is a hyperbolic automorphism or a parabolic automorphism. Further, we show that, in both of the exceptional cases, $$\{C_\varphi \}^{\prime \prime }$${Cφ}″ is the closure of the algebra generated by $$C_\varphi $$Cφ and $$C_\varphi ^{-1},$$Cφ-1, either in the weak operator topology, if $$\varphi $$φ is a hyperbolic automorphism, or surprisingly, in the uniform operator topology, if $$\varphi $$φ is a parabolic automorphism. Finally, for each type of a linear fractional mapping, we settle the question when any of the algebras involved are equal. |
| Starting Page | 501 |
| Ending Page | 532 |
| Page Count | 32 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s13348-019-00244-7 |
| Volume Number | 70 |
| Alternate Webpage(s) | http://homepages.wmich.edu/~petrovic/research/DCP%20final.pdf |
| Alternate Webpage(s) | https://doi.org/10.1007/s13348-019-00244-7 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
